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Abstract

Polyyne and cumulene of infinite length as the typical covalent one-dimensional (1D)
monatomic linear chains of carbon have been demonstrated to be metallic and semiconductor
(Eg = 1.859 eV), respectively, by first-principles calculations. Comparing with single-walled
carbon nanotubes, the densities are evidently low and the thermodynamic properties
are similar below room temperature but much different at the high temperature range.
Polyyne possesses a Young's modulus as high as 1.304 TPa, which means it is even much
stiffer than carbon nanotubes and to be the superlative strong 1D material along the
axial direction. The Young's modulus of cumulene is estimated to be 760.78 GPa. In
addition, polyyne is predicted to be as a one-dimensional electronic material with
very high mobility.

Keywords:

Introduction

Carbon nanomaterials possess different mechanical and electronic properties when the
hybridization of carbon atoms changes from sp, sp2 to sp3. As quasi one-dimensional carbon nanostructures, carbon nanotubes [1], composed of sp2/sp3-hybridized carbon atoms, possess outstanding mechanical, optical, and electronic
properties, offering carbon nanotubes many promising applications as composite materials
and sensors, as well as technology areas such as nanoelectronics. Graphene [2], a one-atom-thick two-dimensional sheet of sp2-hybridized carbon atoms, has also attracted increasing attentions due to its excellent
thermal conductivity, mechanical stiffness, and extraordinary electronic transport
properties, rendering graphene potentially valuable for applications in nanoelectronics,
solar cell window layers, and composite materials [3]. Among all carbon nanostructures, carbon atomic chain, comprised of monatomic linear
chains of carbon atoms, is a truly one-dimensional nanomaterial and has been predicted
to have extremely outstanding properties for various applications [4-6]. Recently, Zhao et al reported the experimental fabrication of monatomic carbon linear
chain which was shown to be inside double-walled carbon nanotubes of 0.7 nm in diameter
[4]. Three characteristic Raman shift peaks were observed in the range of 1,790 to 1,860
cm-1. Mikhailovskij et al. reported fabrication of free-standing short monatomic carbon
linear chains using a developed experimental high-field technique [7]. However, desirable quantity of free-standing polyyne and cumulene samples for experimental
characterization has not gotten until now.

Among various possible covalent one-dimensional (1D) monatomic linear chains of carbon,
cumulene of infinite length (···C ≡ C - C···) constructed with alternate sp conjugated
band and sp3 conjugated bond chain and polyyne of infinite length (···C = C = C···) with sp2 conjugated bond chain are typical rigid covalent period conjugated bond chains. Polyynes
are found in interstellar molecular clouds where hydrogen is scarce. The longest reported
(synthetic) polyyne to date contains 22 acetylenic units and is end-capped with triisopropylsilyl
groups [8]. The polyyne of infinite length is the elusive compound called carbyne; linear acetylenic
carbon is one of the carbon allotropes [9]. To date, the difference of physical properties in these two types of carbon atomic
chains has not been reported. However, there is no effective method reported in the
literature to synthesize and characterize free-standing carbon atomic chains. In this
investigation, theoretical study of carbon atomic chains properties has been conducted.
Two kinds of carbon atomic chains have been investigated by first-principles calculations.
Periodic carbon atomic chain structure has been created and geometry optimized in
order to build a stable structure. Thermodynamic, mechanical, and electronic properties
of carbon atomic chains have been calculated, and the results of physical properties
are discussed.

Calculations

First-principles calculations have been performed to investigate the physical properties
of carbon atomic chains by the pseudo-potential plane wave method [10] with the generalized gradient approximation [11] of Perdew-Burke-Ernzerhof for the exchange-correlation potential [12].

For all structures, the Monkhorst-Pack scheme is used in the Brillouin zone with 2
× 2 × 10 for all the geometry optimization and total energy calculations [13]. For all relaxation calculations, the volume of cell is fixed and the structure considered
are fully relaxed to accuracy where the self-consistent field was done with a convergence
criterion of 2 × 10-5 eV/atom.

Since experiment data on physical properties of single-walled carbon nanotubes (SWNTs)
have been extensively reported [14], the model of a (5,5) SWNT is chosen as a reference for first-principles calculation:
the physical data are compared in order to confirm the accuracy of calculation method
and the feasibility of carbon atomic chain.

With the reference of the distance between two neighbor layers in graphite is 3.42
Å, and linear monatomic carbon linear chain was shown to be inside double-walled carbon
nanotubes of 0.7 nm in diameter, therefore, the smallest distance can be chosen as
3.42 Å between two free carbon atomic chains. The geometrical structure model of (5,5)
SWNT, cumulene, and polyyne are shown in Figure 1.

The calculated geometrical model of (5,5) SWNT contains 80 carbon atoms, as shown
in Figure 1a, with periodic border condition. Structural optimization has been established using
a bond length of 1.42 Å, which is slight shorter than that in graphene inducing from
the surface curving of the SWNT. The model of cumulene as shown in Figure 1b is built with 12 carbon atoms with alternate sp conjugated band and sp3 bonds and periodic border condition. The length of sp conjugated bond is 0.1264 nm
and that of sp3 bond is 0.1476 nm. Polyyne is in the same set as cumulene, as shown in Figure 1c with 12 carbon atoms, and there are only sp2 conjugated bond of length up to 0.1404 nm with periodic border condition.

Results and discussion

The calculated data of (5,5) SWNT, cumulene, and polyyne are listed in Table 1 in which diamond and graphene are included for comparing. At first, the structure
stability of cumulene and polyyne is demonstrated to be lower than SWNT because their
Gibbs free energy is more minus than that of (5,5) SWNT. Secondly, SWNT has been known
as a super light one-dimensional material. However, cumulene and polyyne have even
much lower density for potential applications as typical one-dimensional carbon materials.

It has been reported that the experimental Young's modulus of (5,5) SWNT is about
0.3 to 0.6 TPa with zero bandgap [15,16]. The calculated Young's modulus of SWNTs in this investigation is 805.12 GPa, which
is quite close to the experiment data, demonstrating the successful application of
the first-principles theory to calculate the mechanical properties of carbon nanoscale
structures. The mechanical properties and bandgap energy of (5,5) SWNTs and carbon
atomic chains are listed in Table 1.

The calculated Young's modulus of polyyne is as high as 1,304.19 GPa, which is much
larger than that of (5,5) SWNTs, cumulene (760.78 GPa), or any known materials. The
high Young's modulus of polyyne can be attributed to the high strength of sp2 conjugated bonds. We note, however, that in the SWNT structure, the carbon bonds may
be considered as a hybrid spx (2 < x < 3) bond, which is much weaker than sp2 conjugated bond. In the cumulene structure, there exists the single sp3 bond, which is even weaker than the spx (2 < x < 3) bond in SWNT structure. Therefore, we conclude that polyyne possesses the highest
strength along its axial compared to any other materials.

The calculation results show that cumulene is a semiconductor (Eg = 1.859 eV), but
polyyne is metallic. As we know, the electronic conductivity of carbon structures
comes from enlarged moving range of electrons through sp2 conjugated bonds, which decrease the system energy level and result in the effect
of the electron mobility increasing significantly [17-20]. Because of a single sp3 carbon bond in the cumulene structure, the electronic conductivity is much lower than
that of polyyne, and the bandgap energy of cumulene widens and renders cumulene to
be a semiconductor.

The specific heat capacity chart of cumulene in Figure 2 shows that heat capacity increases with increasing temperature. According to the
thermal equation:

where K is the coefficient of heat conductivity, c is specific heat capacity, m is mass, and X replaces other parameters. Specific heat capacity is proportional to heat capacity;
therefore, the coefficient of heat conductivity is proportional to heat capacity [21]. It can be noticed that the heat conductivity coefficient of cumulene increases with
rising temperature.

SWNT is a one-dimensional super low-density material, with very high Young's modulus.
In this investigation, the heat capacity chart is calculated and shows a trend rises
with increasing temperature gradually, as shown in Figure 2. Comparing at 300 K, the specific heat capacities of cumulene, polyyne, and (5,5)
SWNT are about 1.307, 1.162, and 0.174 J g-1 K-1, respectively. The specific heat capacity of (5,5) SWNT is about an order smaller
than that of the cumulene and polyyne. Meanwhile, the increasing rate of the specific
heat capacity of (5,5) SWNT is slower grading as the temperature increases from 300
K. These results indicate that the thermodynamic properties of cumulene and polyyne
are different obviously from that of SWNTs. SWNT has been known as a super one-dimensional
material with very high Young's modulus and low density for application in many fields.
However, cumulene and polyyne have even much lower density and extremely high Young's
modulus for some potential applications.

On the other hand, low-dimensional carbon materials, such as carbon nanotubes and
graphene, have been reported to have very high mobility values of 100,000 cm2/V·s [22] and 200,000 cm2/V·s [23] at room temperature, respectively. The reason for such high mobility is related to
the carbon structure of spx conjugation. For example, graphite with similar carbon structure is a good conductor
of electricity. The mobility of charges in graphite is as high as 20,000 cm2/V·s at room temperature. Graphite is also strong, lightweight, and an excellent conductor
of heat. The mobilities in nanotubes and graphenes turn out to be even higher than
in graphite because low-dimensional carbon structures of spx conjugation behave as high-mobility electron gases. Another reason may lie in the
quantum confinement nature of the low-dimensional carbon structure, which is harder
to scatter electrons in low-dimensional confined structure. According to the typical
structure of polyyne, it has the most favorable quantum confined carbon structure
of spx conjugation for possessing a possible super high electron mobility and super conductivity
as well. However, cumulene may have a relatively lower electron mobility due to its
sp conjugated band and sp3 bond alternate chain structure. Electrons can only go forward or backward but not
sideways. Metallic carbon atomic chains can behave as one-dimensional electron gases,
having outstanding properties in bandgap energy as well as Young's modulus. Indeed,
the results of this investigation suggest that monatomic carbon linear chains, cumulene
and polyyne, have promising potential application in many fields.

Conclusions

First-principles calculations have been performed to investigate the properties of
covalent one-dimensional (1D) monatomic linear chains of carbon. Polyyne and cumulene
of infinite length, as the two typical chains, have been studied with SWNT, graphene,
and diamond to reveal the fundamental and technologically interesting features. The
Young's modulus of polyyne has been demonstrated to be high as 1.304 TPa, which is
even much higher than that of (5,5) SWNT (0.805 TPa) and diamond (1.050 TPa). This
means that polyyne is an extremely strong 1D material along the axial direction. Specific
heat capacity analysis shows that the thermodynamic properties of cumulene and polyyne
are different obviously from those of SWNTs and that cumulene and polyyne have much
larger specific heat capacity. According to the typical electronic structure of polyyne,
a super high electron mobility is predicted to exist in the most favorable quantum
confined electron gas structure in 1D. Cumulenes as a semiconductor (Eg ≈ 1.859 eV)
and polyynes as a metallic monatomic chains are distinguished from other organic chains
by their rigidity, which makes them promising for applications in nanotechnology.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

YZ carried out the study and drafted the manuscript. YS participated in the sequence
alignment. LW participated in the design of the study and sequence alignment. ESK
participated in the sequence alignment. XC participated in the sequence alignment.
YZ participated in the design of the study and sequence alignment.

Acknowledgements

We acknowledge the financial supports from National Natural Science Foundation of
China (no. 50730008, 61006002), Shanghai Science and Technology Grant (no. 1052nm02000),
and the support from the U-M/SJTU Collaborative Research Program in Renewable Energy
Science and Technology.